Zero Dynamics and Stabilization for Linear DAEs
We study linear differential-algebraic multi-input multi-output systems which are not necessarily regular and investigate the asymptotic stability of the zero dynamics and stabilizability. To this end, the concepts of autonomous zero dynamics, transmission zeros, right-invertibility, stabilizability in the behavioral sense and detectability in the behavioral sense are introduced and algebraic characterizations are derived. It is then proved, for the class of right-invertible systems with autonomous zero dynamics, that asymptotic stability of the zero dynamics is equivalent to three conditions: stabilizability in the behavioral sense, detectability in the behavioral sense, and the condition that all transmission zeros of the system are in the open left complex half-plane. Furthermore, for the same class, it is shown that we can achieve, by a compatible control in the behavioral sense, that the Lyapunov exponent of the interconnected system equals the Lyapunov exponent of the zero dynamics.
KeywordsDifferential-algebraic equations Zero dynamics Transmission zeros Right-invertibility Stabilizability Detectability Lyapunov exponent
I am indebted to Achim Ilchmann (Ilmenau University of Technology) for several constructive discussions.
- 1.Adams, R.A.: Sobolev Spaces. No. 65 in Pure and Applied Mathematics. Academic, New York/London (1975)Google Scholar
- 4.Berger, T.: On differential-algebraic control systems. Ph.D. thesis, Institut für Mathematik, Technische Universität Ilmenau, Universitätsverlag Ilmenau, Ilmenau (2013)Google Scholar
- 5.Berger, T.: Zero dynamics and funnel control of general linear differential-algebraic systems (2013). Submitted for publication. Preprint available from the website of the authorGoogle Scholar
- 6.Berger, T., Ilchmann, A., Reis, T.: Normal forms, high-gain, and funnel control for linear differential-algebraic systems. In: Biegler, L.T., Campbell, S.L., Mehrmann, V. (eds.) Control and Optimization with Differential-Algebraic Constraints. Advances in Design and Control, vol. 23, pp. 127–164. SIAM, Philadelphia (2012)CrossRefGoogle Scholar
- 12.Campbell, S.L., Kunkel, P., Mehrmann, V.: Regularization of linear and nonlinear descriptor systems. In: Biegler, L.T., Campbell, S.L., Mehrmann, V. (eds.) Control and Optimization with Differential-Algebraic Constraints. Advances in Design and Control, vol. 23, pp. 17–36. SIAM, Philadelphia (2012)CrossRefGoogle Scholar
- 15.Isidori, A.: Nonlinear Control Systems. Communications and Control Engineering Series, 3rd edn. Springer, Berlin (1995)Google Scholar